For $t \in(0,2 \pi)$, if $ABC$ is an equilateral triangle with vertices$A ( \sin t, -{\cos t}), B ( \cos t, \sin t )$ and $C ( a , b )$ such that its orthocentre lies on a circle with centre $\left(1, \frac{1}{3}\right)$, then $\left(a^2-b^2\right)$ is equal to :
- $\frac{8}{3}$
- 8
- $\frac{77}{9}$
- $\frac{80}{9}$
The Correct Option is B
Solution and Explanation
From the given options the correct answer is option (B): 8
Top Questions on coordinates of a point in space
- Let $P\left(a_1, b_1\right)$ and $Q\left(a_2, b_2\right)$ be two distinct points on a circle with center $C(\sqrt{2}, \sqrt{3})$ Let $O$ be the origin and $OC$ be perpendicular to both $CP$ and $CQ$. If the area of the triangle $OCP$ is $\frac{\sqrt{35}}{2}$, then $a_1^2+a_2^2+b_1^2+b_2^2$ is equal to ______
- JEE Main - 2023
- Mathematics
- coordinates of a point in space
- The foot of perpendicular from the origin $O$ to a plane $P$ which meets the co-ordinate axes at the points $A , B , C$ is $(2, a , 4), a \in N$ If the volume of the tetrahedron $OABC$ is 144 unit $^3$, then which of the following points is NOT on $P$ ?
- JEE Main - 2023
- Mathematics
- coordinates of a point in space
- The area of the quadrilateral having vertices as (1,2), (5,6), (7,6), (-1,-6) is?
- JEE Main - 2023
- Mathematics
- coordinates of a point in space
- \(\lim_{{y \to 0}} \frac{\sqrt{3} + y^3 - \sqrt{3}}{y^3}\)
- KCET - 2022
- Mathematics
- coordinates of a point in space
- The octant in which the point (2,-4,-7) lies is
- KCET - 2022
- Mathematics
- coordinates of a point in space
Questions Asked in JEE Main exam
- A convex lens of focal length 40 cm forms an image of an extended source of light on a photo-electric cell. A current \( I \) is produced. The lens is replaced by another convex lens having the same diameter but focal length 20 cm. The photoelectric current now is:
- JEE Main - 2024
- Dual nature of matter
- For a sparingly soluble salt \( \text{AB}_2 \), the equilibrium concentrations of \( \text{A}^{2+} \) ions and \( \text{B}^- \) ions are \( 1.2 \times 10^{-4} \, \text{M} \) and \( 0.24 \times 10^{-3} \, \text{M} \), respectively. The solubility product of \( \text{AB}_2 \) is:
- JEE Main - 2024
- Method of Preparation of Diazoniun Salts
The portion of the line \( 4x + 5y = 20 \) in the first quadrant is trisected by the lines \( L_1 \) and \( L_2 \) passing through the origin. The tangent of an angle between the lines \( L_1 \) and \( L_2 \) is:
- JEE Main - 2024
- Tangents and Normals
- Let \[ \vec{a} = 2\hat{i} + \alpha \hat{j} + \hat{k}, \quad \vec{b} = -\hat{i} + \hat{k}, \quad \vec{c} = \beta \hat{j} - \hat{k}, \] where \( \alpha \) and \( \beta \) are integers and \( \alpha \beta = -6 \). Let the values of the ordered pair \( (\alpha, \beta) \) for which the area of the parallelogram of diagonals \( \vec{a} + \vec{b} \) and \( \vec{b} + \vec{c} \) is \( \frac{\sqrt{21}}{2} \), be \( (\alpha_1, \beta_1) \) and \( (\alpha_2, \beta_2) \). Then \( \alpha_1^2 + \beta_1^2 - \alpha_2 \beta_2 \) is equal to:
- JEE Main - 2024
- Vectors
- The molecular formula of second homologue in the homologous series of monocarboxylic acid is
- JEE Main - 2024
- Aldehydes, Ketones and Carboxylic Acids
Concepts Used:
Coordinates of a Point in Space
Three-dimensional space is also named 3-space or tri-dimensional space.
It is a geometric setting that carries three values needed to set the position of an element. In Mathematics and Physics, a sequence of ‘n’ numbers can be acknowledged as a location in ‘n-dimensional space’. When n = 3 it is named a three-dimensional Euclidean space.
The Distance Formula Between the Two Points in Three Dimension is as follows;
The distance between two points P1 and P2 are (x1, y1) and (x2, y2) respectively in the XY-plane is expressed by the distance formula,
Read More: Coordinates of a Point in Three Dimensions